## Probability in the North East workshop

#### 4 May 2022

Organizers: Martín López-García and Matthew Aldridge (Leeds).

Venue: Roger Stevens LT 06, University of Leeds

Attendence is free but registration is required for organisational purposes, by emailing Martín López-García by Thursday 28th April 2022, also mentioning any special dietary requirements you may have.

## Programme

12:30–13:20
Lunch
13:20–14:10
Bevelynn Williams (University of Leeds)
During inhalational anthrax infection, Bacillus anthracis spores are ingested by phagocytes such as alveolar macrophages and dendritic cells. The spores begin to germinate and then proliferate inside the phagocytes, which may eventually lead to death of the host cell and the release of bacteria into the extracellular environment. Alternatively, some phagocytes may be successful in eliminating the intracellular bacteria and will recover. We consider a stochastic, Markov chain model for the intracellular infection dynamics of B. anthracis in a single phagocyte, incorporating spore germination and maturation, bacterial proliferation and death, and the possible release of bacteria due to cell rupture. From this model, we can derive the probability distribution for the rupture size of infected phagocytes, as well as the distribution of time until phagocyte rupture and bacterial release. The model accounts for potential heterogeneity in the spore germination rate, with the consideration of two extreme cases for the rate distribution: continuous Gaussian and discrete Bernoulli. Through Bayesian inference, the model is parameterised using in vitro measurements of intracellular spore and bacterial counts. Our results support the existence of significant heterogeneity in the germination rate across different spores, with a subset of spores expected to germinate much later than the majority. Furthermore, in agreement with experimental evidence, our results suggest that the majority of spores taken up by macrophages are likely to be eliminated by the host cell, but a few germinated spores may survive phagocytosis and lead to the death of the infected cell. Finally, we discuss how this stochastic modelling approach, together with dose-response data, can allow us to quantify and predict individual infection risk following exposure.
14:10–15:00
Theo Assiotis (University of Edinburgh)
The Laguerre-Polya class of entire functions consists of all functions arising as uniform limits of polynomials with only real roots. In this talk I will discuss how any random Laguerre-Polya function can be obtained from random unitarily invariant infinite Hermitian matrices via a natural scaling limit of the corresponding characteristic polynomial. For some distinguished cases of random Laguerre-Polya functions the first Taylor coefficient is known to be connected to integrable systems.
15:00–15:30
Lunch
15:30–16:20
Huy Chau (University of Manchester)
This talk is devoted to robust fundamental theorems of asset pricing in discrete time and finite horizon settings. The new concept "robust pricing system" is introduced to rule out the existence of model independent arbitrage opportunities. Superhedging duality and strategy are obtained.
16:20–17:10
Sijia Li (University of Leeds)
Classically, statistical datasets have a larger number of data points than features (pn such models are poorly determined. Kalaitzis et al. 2013 introduced the Bigraphical Lasso, an estimator for sparse precision matrices based on the Cartesian product of graphs. Unfortunately, the original Bigraphical Lasso algorithm is not applicable in case of large p and n due to memory requirements. We exploit eigenvalue decomposition of the Cartesian product graph to present a more efficient version of the algorithm which reduces memory requirements from $O(n^2 p^2)$ to $O(n^2+p^2)$. Many datasets in different application fields, such as biology, medicine and social science, come with count data, for which Gaussian based models are not applicable. Our multi-way network inference approach can be used for discrete data.

Our methodology accounts for the dependencies across both instances and features, reduces the computational complexity for high dimensional data and enables to deal with both discrete and continuous data. Numerical studies on both synthetic and real datasets are presented to showcase the performance of our method.